For \(\operatorname{Re}(s)>-n\) , \(h_s\)  is locally summable with moderate growth at infinity, hence defines a tempered distribution. The (\(\mathcal {S}^{\prime }(\mathbb {R}^n)\) -valued) function \(s \mapsto h_s\)  can be extended by analytic continuation to \(\mathbb {C}\)  and the \(\Gamma \)  factor in the definition of \(h_s\)  is so chosen that it extends as an entire function with values in \(\mathcal {S}^{\prime }(\mathbb {R}^n)\)  (for more details see, e.g., [1]}).
